Grade 10 Maths · Trigonometry · Solve with two right triangles

Solve with two right triangles Grade 10

Find a side in one right-angled triangle and use it as the shared side of a second triangle, for example in angle of elevation and depression problems.

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Trigonometry 2D Grade 10 Angle of elevation and depressionMiss Martins Maths and Science

Grade 10 Trigonometry Exam Questions PART 2: Angle of elevation and Depression QuestionMiss Martins Maths and Science

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Question 6.3 · November 2024 · Paper 2 · Provincial · 4 marks

An observer on the top of a cliff, 200 m above the sea level, observes the angles of depression of the two ships to be

4 5^{\circ}

for ship A, and

3 0^{\circ}

for ship B. Diagram for this question

6.3 Determine the distance between Ship A and Ship B.(4)

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More questions: solve with two right triangles

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3. Practice · NavyBlue · 2 marks
In $△ A BC$ , $A D ⊥ BC$ with $D$ on $BC$ . $A B = 14$ cm, $\hat{B} = 4 8^{\circ}$ and $\hat{C} = 3 5^{\circ}$ . Give answers correct to two decimal places.
(b) Hence calculate the length of $A C$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  Ratio in triangle ADCIn $△ A D C$ : $sin 3 5^{\circ} = \frac{A D}{A C} = \frac{10 , 40}{A C}$
  1 mark · Solve with two right triangles
2. ✓
  AC $A C = \frac{10 , 40}{s i n 3 5 ^{\circ}} = 18, 14$ cm
  1 mark · Solve with two right triangles
4. Practice · NavyBlue · 2 marks
A building $TF$ is $24$ m high. From the top $T$ of the building, the angle of elevation of the top $P$ of a tower $PQ$ is $2 8^{\circ}$ , and the angle of depression of the foot $Q$ of the tower is $4 0^{\circ}$ . The building and the tower stand on level ground. $TN$ is horizontal, with $N$ on $PQ$ .
(b) Calculate the height of the tower $PQ$ .(2)
Show the memo (2 ticks)Memo
1. ✓
  PN $TN = FQ = 28, 60$ m
  In $△ TNP$ : $PN = 28, 60 \times tan 2 8^{\circ} = 15, 21$ m
  1 mark · Solve with two right triangles
2. ✓
  PQ $PQ = PN + NQ = 15, 21 + 24 = 39, 21$ m
  1 mark · Solve with two right triangles
5. Practice · Siyavula Grade 10 Maths · Ex 11.2 · 4 marks
When the top, $T$ , of a mountain is viewed from point $A$ , $2000$ m from the ground, the angle of depression is equal to $1 5^{\circ}$ . When it is viewed from point $B$ on the ground, the angle of elevation is equal to $1 0^{\circ}$ . If the points $A$ and $B$ are on the same vertical line, find the height, $h$ , of the mountain. Round your answer to one decimal place.(4)
Show the memo (4 ticks)Memo
1. ✓
  Upper triangleIn $△ A CT$ : $tan 1 5^{\circ} = \frac{2000 - h}{d}$ , so $2000 - h = d tan 1 5^{\circ}$
  1 mark · Calculate a side using trig
2. ✓
  Lower triangleIn $△ BCT$ : $tan 1 0^{\circ} = \frac{h}{d}$ , so $h = d tan 1 0^{\circ}$
  1 mark · Calculate a side using trig
3. ✓
  Find the shared sideAdd: $2000 = d tan 1 5^{\circ} + d tan 1 0^{\circ} = d (tan 1 5^{\circ} + tan 1 0^{\circ})$
  $d = \frac{2000}{t a n 1 5 ^{\circ} + t a n 1 0 ^{\circ}} = 4501, 70$ m
  1 mark · Solve with two right triangles
4. ✓
  Height $h = 4501, 70 \times tan 1 0^{\circ} = 793, 8$ m
  1 mark · Solve with two right triangles
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue
6. Practice · Siyavula Grade 10 Maths · Example 4 · 4 marks
A block of flats is $200$ m away from a cellphone tower. Someone stands at $B$ . They measure the angle from $B$ to the top of the tower ( $E$ ) to be $3 4^{\circ}$ (the angle of elevation). They then measure the angle from $B$ to the bottom of the tower ( $C$ ) to be $6 2^{\circ}$ (the angle of depression). What is the height of the cellphone tower (correct to the nearest metre)?(4)
Show the memo (4 ticks)Memo
1. ✓
  Shared side $A B D C$ is a rectangle, so $B D = A C = 200$ m.
  1 mark · Solve with two right triangles
2. ✓
  CDIn $△ B D C$ : $tan 6 2^{\circ} = \frac{C D}{200}$
  $C D = 200 tan 6 2^{\circ} = 376, 15$ m
  1 mark · Calculate a side using trig
3. ✓
  DEIn $△ B D E$ : $tan 3 4^{\circ} = \frac{D E}{200}$
  $D E = 200 tan 3 4^{\circ} = 134, 90$ m
  1 mark · Calculate a side using trig
4. ✓
  Height of tower $CE = C D + D E = 376, 15 + 134, 90 = 511, 05 \approx 511$ m
  1 mark · Solve with two right triangles
Question adapted from Siyavula Education (CC BY 3.0) · Solution by NavyBlue

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